Tsallis 70 Complex Systems Foundations and Applications
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On Escort Distributions, Q-Gaussians and Fisher Information Jean-François Bercher
On escort distributions, q-gaussians and Fisher information Jean-François Bercher To cite this version: Jean-François Bercher. On escort distributions, q-gaussians and Fisher information. 30th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Jul 2010, Chamonix, France. pp.208-215, 10.1063/1.3573618. hal-00766571 HAL Id: hal-00766571 https://hal-upec-upem.archives-ouvertes.fr/hal-00766571 Submitted on 18 Dec 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On escort distributions, q-gaussians and Fisher information J.-F. Bercher Laboratoire d’Informatique Gaspard Monge, Université Paris-Est, ESIEE, 5 bd Descartes, 77454 Marne-la-Vallée Cedex 2, France Abstract. Escort distributions are a simple one parameter deformation of an original distribution p. In Tsallis extended thermostatistics, the escort-averages, defined with respect to an escort dis- tribution, have revealed useful in order to obtain analytical results and variational equations, with in particular the equilibrium distributions obtained as maxima of Rényi-Tsallis entropy subject to constraints in the form of a q-average. A central example is the q-gaussian, which is a generalization of the standard gaussian distribution. -
Luis Gregorio Moyano
Luis Gregorio Moyano Rep´ublicadel L´ıbano 150, Mendoza, 5500 Mendoza, Argentina [email protected] Current position Facultad de Ciencias Exactas y Naturales Mendoza, Argentina Universidad Nacional de Cuyo Adjunt Professor May 2016 - Today CONICET Adjunt Researcher May 2016 - Today Education and training Universidad Carlos III de Madrid Madrid, Spain Postdoctoral Position at 2007 - 2008 Mathematics Department & GISC Centro Brasileiro de Pesquisas F´ısicas Rio de Janeiro, Brazil PhD in Physics 2001 - 2006 Supervisor: Prof. Constantino Tsallis. PhD Thesis: \Nonextensive statistical mechanics in complex systems: dynamical foundations and applications" Instituto Balseiro Bariloche, Argentina MSc in Physics 1997- 2000 Supervisors: Prof. Dami´anZanette and Prof. Guillermo Abramson. Msc. Thesis: \Learning in coupled dynamical systems" Research Interests • Machine Learning applications on networks, Natural Language Processing, Big Data applications. • Complex Networks, Social Networks, Complex Systems. • Statistical mechanics, applications to economical, social and biological systems. • Bayesian inference, Markovian systems, congestion and information transfer dynamics in networks. • Evolutionary game theory, emergence of cooperation in complex networks. • Nonlinear dynamics, coupled chaotic systems, synchronization. Luis G. Moyano - Curriculum Vitæ 1 Positions, fellowships and awards • IBM Research Brazil, Rio de Janeiro, Brazil - Staff Research Member December, 2013 - April, 2016 Staff Research Member at the Social Data Analytics -
Associating an Entropy with Power-Law Frequency of Events
entropy Article Associating an Entropy with Power-Law Frequency of Events Evaldo M. F. Curado 1, Fernando D. Nobre 1,* and Angel Plastino 2 1 Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil; [email protected] 2 La Plata National University and Argentina’s National Research Council (IFLP-CCT-CONICET)-C. C. 727, La Plata 1900, Argentina; plastino@fisica.unlp.edu.ar * Correspondence: [email protected]; Tel.: +55-21-2141-7513 Received: 3 October 2018; Accepted: 23 November 2018; Published: 6 December 2018 Abstract: Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy Sq. The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy #0, in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature Tq with kTq ∝ #0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of Tq are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures. -
Tsallis Statistics of the Magnetic Field in the Heliosheath Lf Burlaga,1 Af Vin
The Astrophysical Journal, 644:L83–L86, 2006 June 10 ᭧ 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A. TSALLIS STATISTICS OF THE MAGNETIC FIELD IN THE HELIOSHEATH L. F. Burlaga,1 A. F. Vin˜ as,1 N. F. Ness,2 and M. H. Acun˜a3 Received 2006 March 27; accepted 2006 May 3; published 2006 May 25 ABSTRACT The spacecraft Voyager 1 crossed the termination shock on 2004 December 16 at a distance of 94 AU from the Sun, and it has been moving through the heliosheath toward the interstellar medium since then. The distributions of magnetic field strength B observed in the heliosheath are Gaussian over a wide range of scales, yet the measured profile appears to be filamentary with occasional large jumps in the magnetic field strengthB(t) . All of the probability distributions of changes in B,dBn { B(t ϩ t) Ϫ B(t) on scales t from 1 to 128 days, can be fit with the symmetric Tsallis distribution of nonextensive statistical mechanics. At scales ≥32 days, the distributions are Gaussian, but on scales from 1 through 16 days the probability distribution functions have non-Gaussian tails, suggesting that the inner heliosheath is not in statistical equilibrium on scales from 1 to 16 days. Subject headings: magnetic fields — solar wind — turbulence — waves 1. INTRODUCTION is expressed by the form of u in the distribution function. In The solar wind moves supersonically beyond a few solar Tsallis statistical mechanics, the probability that a system is in a state with energy u is proportional toexp (Ϫ u) . -
The Entropy Universe
entropy Review The Entropy Universe Maria Ribeiro 1,2,* , Teresa Henriques 3,4 , Luísa Castro 3 , André Souto 5,6,7 , Luís Antunes 1,2 , Cristina Costa-Santos 3,4 and Andreia Teixeira 3,4,8 1 Institute for Systems and Computer Engineering, Technology and Science (INESC-TEC), 4200-465 Porto, Portugal; [email protected] 2 Computer Science Department, Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal 3 Centre for Health Technology and Services Research (CINTESIS), Faculty of Medicine University of Porto, 4200-450 Porto, Portugal; [email protected] (T.H.); [email protected] (L.C.); [email protected] (C.C.-S.); andreiasofi[email protected] (A.T.) 4 Department of Community Medicine, Information and Health Decision Sciences-MEDCIDS, Faculty of Medicine, University of Porto, 4200-450 Porto, Portugal 5 LASIGE, Faculdade de Ciências da Universidade de Lisboa, 1749-016 Lisboa, Portugal; [email protected] 6 Departamento de Informática, Faculdade de Ciências da Universidade de Lisboa, 1749-016 Lisboa, Portugal 7 Instituto de Telecomunicações, 1049-001 Lisboa, Portugal 8 Instituto Politécnico de Viana do Castelo, 4900-347 Viana do Castelo, Portugal * Correspondence: [email protected] Abstract: About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical linguistics. This paper presents The Entropy -
Arxiv:1707.03526V1 [Cond-Mat.Stat-Mech] 12 Jul 2017 Eq
Generalized Ensemble Theory with Non-extensive Statistics Ke-Ming Shen,∗ Ben-Wei Zhang,y and En-Ke Wang Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China (Dated: October 17, 2018) The non-extensive canonical ensemble theory is reconsidered with the method of Lagrange multipliers by maximizing Tsallis entropy, with the constraint that the normalized term of P q Tsallis' q−average of physical quantities, the sum pj , is independent of the probability pi for Tsallis parameter q. The self-referential problem in the deduced probability and thermal quantities in non-extensive statistics is thus avoided, and thermodynamical relationships are obtained in a consistent and natural way. We also extend the study to the non-extensive grand canonical ensemble theory and obtain the q-deformed Bose-Einstein distribution as well as the q-deformed Fermi-Dirac distribution. The theory is further applied to the general- ized Planck law to demonstrate the distinct behaviors of the various generalized q-distribution functions discussed in literature. I. INTRODUCTION In the last thirty years the non-extensive statistical mechanics, based on Tsallis entropy [1,2] and the corresponding deformed exponential function, has been developed and attracted a lot of attentions with a large amount of applications in rather diversified fields [3]. Tsallis non- extensive statistical mechanics is a generalization of the common Boltzmann-Gibbs (BG) statistical PW mechanics by postulating a generalized entropy of the classical one, S = −k i=1 pi ln pi: W X q Sq = −k pi lnq pi ; (1) i=1 where k is a positive constant and denotes Boltzmann constant in BG statistical mechanics. -
Beyond Behavioural Economics
Beyond Behavioural Economics A Process Realist Perspective Mathematics of Behavioural Economics and Finance James Juniper November, 2015 Ultimate scientific explanation is attractive and important, but financial engineering cannot wait for full explanation. That is, it is legitimate to strive towards a second best: a “descriptive phenomenology” that is organized tightly enough to bring a degree of order and understanding. Benoit B. Mandelbrot (1997) Overview • Keynes on DM under uncertainty & role of conventions • Ontological considerations • Objective not just subjective • Applications: • Bounded sub-additivity (Tversky & Wakker) • Equivalent approach—fuzzy measure theory (approximation algebras) • Concept lattices and “framing” • Tsallis arithmetic: q-generalized binomial approximation […] civilization is a thin and precarious crust erected by the personality and will of a very few and only maintained by rules and conventions skillfully put across and guilefully preserved” (Keynes, X: 447) The “love of money as a possession” is a “somewhat disgusting morbidity, one of those semi-criminal, semi-pathological propensities which one hands over with a shudder to the specialists in mental disease (IX: 329) For we shall enquire more curiously than is safe today into the true character of this ‘purposiveness’ with which in varying degrees Nature has endowed almost all of us. For purposiveness means that we are more concerned with the remote future results of our actions than with their own quality or their immediate effects on the environment. The ‘purposive’ man is always trying to secure a spurious and delusive immortality for his acts by pushing his interest in them forward into time. He does not love his cat, but his cat’s kittens; nor, in truth, the kittens, but only the kittens’ kittens, and so on forward to the end of catdom. -
Nonextensive Statistical Mechanics: a Brief Review of Its Present Status
Anais da Academia Brasileira de Ciências (2002) 74(3): 393–414 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc Nonextensive statistical mechanics: a brief review of its present status CONSTANTINO TSALLIS* Centro Brasileiro de Pesquisas Físicas, 22290-180 Rio de Janeiro, Brazil Centro de Fisica da Materia Condensada, Universidade de Lisboa P-1649-003 Lisboa, Portugal Manuscript received on May 25, 2002; accepted for publication on May 27, 2002. ABSTRACT We briefly review the present status of nonextensive statistical mechanics. We focus on (i) the cen- tral equations of the formalism, (ii) the most recent applications in physics and other sciences, (iii) the a priori determination (from microscopic dynamics) of the entropic index q for two important classes of physical systems, namely low-dimensional maps (both dissipative and conservative) and long-range interacting many-body hamiltonian classical systems. Key words: nonextensive statistical mechanics, entropy, complex systems. 1 CENTRAL EQUATIONS OF NONEXTENSIVE STATISTICAL MECHANICS Nonextensive statistical mechanics and thermodynamics were introduced in 1988 [1], and further developed in 1991 [2] and 1998 [3], with the aim of extending the domain of applicability of statistical mechanical procedures to systems where Boltzmann-Gibbs (BG) thermal statistics and standard thermodynamics present serious mathematical difficulties or just plainly fail. Indeed, a rapidly increasing number of systems are pointed out in the literature for which the usual functions appearing in BG statistics appear to be violated. Some of these cases are satisfactorily handled within the formalism we are here addressing (see [4] for reviews and [5] for a regularly updated bibliography which includes crucial contributions and clarifications that many scientists have given along the years). -
Fractal Structure and Non-Extensive Statistics
entropy Article Fractal Structure and Non-Extensive Statistics Airton Deppman 1,* ID , Tobias Frederico 2, Eugenio Megías 3,4 and Debora P. Menezes 5 1 Instituto de Física, Universidade de São Paulo, Rua do Matão Travessa R Nr.187, Cidade Universitária, CEP 05508-090 São Paulo, Brazil 2 Instituto Tecnológico da Aeronáutica, 12228-900 São José dos Campos, Brazil; [email protected] 3 Departamento de Física Teórica, Universidad del País Vasco UPV/EHU, Apartado 644, 48080 Bilbao, Spain; [email protected] 4 Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Avenida de Fuente Nueva s/n, 18071 Granada, Spain 5 Departamento de Física, CFM, Universidade Federal de Santa Catarina, CP 476, CEP 88040-900 Florianópolis, Brazil; [email protected] * Correspondence: [email protected] Received: 27 June 2018; Accepted: 19 August 2018; Published: 24 August 2018 Abstract: The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. -
Thermodynamics from First Principles: Correlations and Nonextensivity
Thermodynamics from first principles: correlations and nonextensivity S. N. Saadatmand,1, ∗ Tim Gould,2 E. G. Cavalcanti,3 and J. A. Vaccaro1 1Centre for Quantum Dynamics, Griffith University, Nathan, QLD 4111, Australia. 2Qld Micro- and Nanotechnology Centre, Griffith University, Nathan, QLD 4111, Australia. 3Centre for Quantum Dynamics, Griffith University, Gold Coast, QLD 4222, Australia. (Dated: March 17, 2020) The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this letter, we use the fundamental principles of ergodicity (via Liouville's theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Boltzmann-Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics. Introduction. The ability to describe the statistical tion [30] or the structure of the microstates [27]. Another state of a macroscopic system is central to many ar- widely-used approach is to generalize the MaxEnt prin- eas of physics [1{4]. In thermostatistics, the statistical ciple to apply to a different entropy functional in place of BGS state of a system of N particles in equilibrium is de- S (fwzg). -
Q-Gaussian Approximants Mimic Non-Extensive Statistical-Mechanical Expectation for Many-Body Probabilistic Model with Long-Range Correlations
Cent. Eur. J. Phys. • 7(3) • 2009 • 387-394 DOI: 10.2478/s11534-009-0054-4 Central European Journal of Physics q-Gaussian approximants mimic non-extensive statistical-mechanical expectation for many-body probabilistic model with long-range correlations Research Article William J. Thistleton1∗, John A. Marsh2† , Kenric P. Nelson3‡ , Constantino Tsallis45§ 1 Department of Mathematics, SUNY Institute of Technology, Utica NY 13504, USA 2 Department of Computer and Information Sciences, SUNY Institute of Technology, Utica NY 13504, USA 3 Raytheon Integrated Defense Systems, Principal Systems Engineer 4 Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro - RJ, Brazil 5 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA Received 3 November 2008; accepted 25 March 2009 Abstract: We study a strictly scale-invariant probabilistic N-body model with symmetric, uniform, identically distributed random variables. Correlations are induced through a transformation of a multivariate Gaussian distribution with covariance matrix decaying out from the unit diagonal, as ρ/rα for r =1, 2, …, N-1, where r indicates displacement from the diagonal and where 0 6 ρ 6 1 and α > 0. We show numerically that the sum of the N dependent random variables is well modeled by a compact support q-Gaussian distribution. In the particular case of α = 0 we obtain q = (1-5/3 ρ) / (1- ρ), a result validated analytically in a recent paper by Hilhorst and Schehr. Our present results with these q-Gaussian approximants precisely mimic the behavior expected in the frame of non-extensive statistical mechanics. -
Some Comments on Boltzmann-Gibbs Statistical Mechanics
CBPF-NF-001/94 Some Comments on Boltzmann-Gibbs Statistical Mechanics Constantino TSALLIS Centro Brasileiro de Pesquisas Fsicas Rua Dr. Xavier Sigaud, 150 22290-180 { Rio de Janeiro { RJ, Brazil Abstract A nonexhaustive review is presented of the limits of the impressive and vastly known success of Boltzmann-Gibbs statistics and normal thermo dynamics. These limits naturally op en the do or for the research of generalized formalisms that could enlarge the domain of validity of standard statistical mechanics and thermo dynam- ics. A p ossible such generalization (recently prop osed by the author) is commented along this p ersp ective. CBPF-NF-001/94 1 1 Limitations of Boltzmann-Gibbs Statistics 1.1 Intro duction The qualitative and quantitative success of Boltzmann-Gibbs (BG) Statistical Mechan- ics (and, naturally, of its particular cases, the Fermi-Dirac and Bose-Einstein quantum statistics, with their common high temp erature asymptotic limit, the classical Maxwell- Boltzmann statistics) is so ubiquitous, persistent and delicate that not few physicists and chemists have a kind of strong (not necessarily rationalized) feeling that this brilliant for- malism is, in practical terms, universal, eternal and in nitely precise. A more balanced analysis reveals some of its sp eci c limitations and inadequacies, and consequently the fragility or nonuniversality of some of the basic hyp othesis of its foundations. Among these foundation stones, a privileged p osition is detained by the entropy S ,asintro duced R and used by Boltzmann and Gibbs (S = k dxf (x)lnf (x) for classical distribution B laws f (x) de ned in phase space), further generalized byvon Neumann (S = k Tr ln , B b eing the density op erator de ned in Hilb ert or Fock spaces), with its diagonal form W (S = k p ln p , p b eing the probability of the i-th among W microstates, and its B i i i i=1 famous equal-probability particular form S = k ln W ) as nely discussed, in the context B of Information Theory,by Shannon.